Acoustics of a partially partitioned narrow slit connected to a half-plane: case study for exponential quasi-bound states in the continuum and their resonant excitation

TL;DR

This paper investigates the asymptotic behavior of quasi-bound states in the continuum in an acoustic system with a narrow slit, deriving formulas for their small decay rates and large resonant amplitudes, validated by numerical results.

Contribution

It provides the first asymptotic analysis of exponentially small decay rates and large resonant amplitudes of quasi-BIC in an acoustic slit system using matched asymptotic expansions.

Findings

Exponential approach of quasi-BIC to true BIC as slit narrows.

Leading approximation for the exponentially small imaginary part of eigenvalues.

Resonant excitation leads to large field enhancements and cylindrical-dipolar waves.

Abstract

Localised wave oscillations in an open system that do not decay or grow in time, despite their frequency lying within a continuous spectrum of radiation modes carrying energy to or from infinity, are known as bound states in the continuum (BIC). Small perturbations from the typically delicate conditions for BIC almost always result in the waves weakly coupling with the radiation modes, leading to leaky states called quasi-BIC that have a large quality factor. We study the asymptotic nature of this weak coupling in the case of acoustic waves interacting with a rigid substrate featuring a partially partitioned slit -- a setup that supports quasi-BIC that exponentially approach BIC as the slit is made increasingly narrow. In that limit, we use the method of matched asymptotic expansions in conjunction with reciprocal relations to study those quasi-BIC and their resonant excitation. In particular, we derive a leading approximation for the exponentially small imaginary part of each wavenumber eigenvalue (inversely proportional to quality factor), which is beyond all orders of the expansion for the wavenumber eigenvalue itself. Furthermore, we derive a leading approximation for the exponentially large amplitudes of the states in the case where they are resonantly excited by a plane wave at oblique incidence. These resonances occur in exponentially narrow wavenumber intervals and are physically manifested in cylindrical-dipolar waves emanating from the slit aperture and exponentially large field enhancements inside the slit. The asymptotic approximations are validated against numerical calculations.